document.write( "Question 1151756: A young wizard knows every third spell that an old wizard knows. However, the
\n" ); document.write( "young wizard knows two spells that the old wizard does not know. 8/9 of all the
\n" ); document.write( "spells that the young wizard knows are known to the old wizard as well. How
\n" ); document.write( "many spells are there that at least one of them knows? \n" ); document.write( "

Algebra.Com's Answer #773619 by ikleyn(52803) $\"\"$ $\"About$

You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " It is good problem from the category \"HOW TO ALGEBREZE it . . . \"\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let  x = number of spells Old wizard knows;\r\n" );
document.write( "\r\n" );
document.write( "     y = number of spells Young wizard knows;\r\n" );
document.write( "\r\n" );
document.write( "     z = number of spells BOTH wizards know (intersection).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then from the condition, you have these equations\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     z = ,               (1)    (A young wizard knows every third spell that an old wizard knows.)\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "     y = ;             (2)    (A young wizard knows every third spell that an old wizard knows. However, the\r\n" );
document.write( "                                           young wizard knows two spells that the old wizard does not know. )\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     z = .               (3)    (8/9 of all the spells that the young wizard knows are known to the old wizard as well.)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus we have three equations for 3 unknowns.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To solve the system, from (1) and (3) you have\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "      = ,\r\n" );
document.write( "\r\n" );
document.write( "which implies\r\n" );
document.write( "\r\n" );
document.write( "     3x = 8y.                 (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From equation (2), substitute  y =   into equation (4).  You will get\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    3x = .          (5)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides of equation (5) by 3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     9x = 8x + 8*2*3,    or\r\n" );
document.write( "\r\n" );
document.write( "     x  = 48.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then from  (1)  z = 16,  and from  (2)  y = 18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now the value under the question is  x + y - z = 48 + 18 - 16 = 50.    ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It also can be solved via logical reasoning, without using equations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The problem says  \"8/9 of all the spells that the young wizard knows are known to the old wizard as well\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It also says that  \"the young wizard knows two spells that the old wizard does not know.\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means that  1/9  of the words young wizard knows is 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, the number of words that young wizard knows is 18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means that the number of words that BOTH wizards know is  18-2 = 16.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In turn, it implies that the number of words the old wizard knows is  3*16 = 48.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "And the answer to the problem question  is  then  48 + 18 - 16 = 50 - the same aa above.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );